Remark 2 (Role of partition \(\alpha \) )  
In our Algorithms REF -REF , the choice of partition \(\alpha \)  brings flexibility in balancing the computational efficiency and solution quality at each iteration. Choosing a suitable partition might be problem dependent; we refer the interested reader to [1]} for more discussions. Here, we highlight that 1) a coarser partition normally leads to a faster convergence in Algorithms REF -REF , as shown in our extensive numerical experiments in Section ; 2) a coarser partition also leads a smaller number \(p\)  in (REF ) and (REF ). The latter fact is important in constructing the problem in each iteration, especially for large-scale cases. For example, when \(n = 2000\) , if we use the SDD matrices for inner/outer approximations [2]}, [3]}, the number of small blocks is \(\binom{2000}{2} = 1\,999\,000\)  which is too large to even construct the problem instances (REF ) and (REF ). We indeed failed to construct such problems in our experiments in Section REF . Instead, for \(\alpha = \lbrace 10,\ldots ,10\rbrace \)  (\(\beta = \lbrace 20,\ldots ,20\rbrace \) , respectively), the number of blocks is reduced to \( \binom{200}{2}=19900\)  (\(\binom{100}{2}=4950\) , respectively), for which efficient constructions exist.

